What is

If we graph the functions x² and 2^x, they intersect in three places:

One of the points of intersection has a negative value of x, and two have positive values of x. After those intersections, as x approaches ∞ the graph of
2^x will always be on top of the function x².

Here are some values of the two functions. We can see that when x = 2 and x = 4 the functions intersect, and that when x is greater than 4 the function 2^x is pulling away from the function x². It's like a horse race, where the function x² is not having a good day.

If we take these values and look at the quotient , here's what happens:

We're already down to about 0.098 when x = 10, and we've barely started! When x = 100, we're down to

Now that's small. We conclude that

What is

We already know these functions intersect for some negative value of x and at x = 2 and x = 4. We also know that the function x² is losing the horse race: for all x greater than 4, x² will be below (or behind) 2^x.

Taking the same values, we'll look at the new ratios now that we've turned the fraction over.

These numbers don't seem to be approaching anything yet. Try something larger:

We find that as x gets larger, the quotient will also get larger and larger without bound. Therefore